15 WEAK TOPOLOGIES 73 15. WEAK TOPOLOGIES (12.15.1) Let (Ea)ae i be a family of locally convex spaces, and for each a let GwKeu be a family of seminorms defining the topology ofEa. Let E = JQ Ea aeE be the product vector space, and for each a el and each X e La, put (220.127.116.11) PU*) = P«(pt*x) far all xeE. Then the p^ are seminorms on E and define the product topology (12.5) on E. It is straightforward to check that the p'^ are seminorms, and then the second assertion follows from the definition (12.5) of a product of uniformiz- able spaces. In particular, when all the Ea are equal to the same locally convex space F, then the product topology on the vector space F1 of all mappings of I into F is defined by the seminorms where a e I and p belongs to a set F of seminorms defining the topology of F. This topology is called the topology of simple (or pointwise) convergence on the vector space F1, or on any of its vector subspaces (cf. (12.5)). With respect to this topojpgy, each mapping/i-»/(a) (a e I) is a continuous linear mapping on F1, and maps each open set in F1 to an open set in F (12.5.2). The topology of simple convergence is Hausdorff if F is Hausdorff (12.5.7). A mapping t\—>ft of a topological space T into F1 is continuous with respect to the topology of simple convergence if and only if the mapping /»->/t(a) of T into F is continuous, for each a 61 (12.5.5). When F is a Banach space over K (=R or C) and T is an open subset of Kn, a mapping t\-*ft of T into F1 is said to be p times differentiate (resp. indefinitely differentiate, resp. analytic) with respect to the topology of simple convergence if, for each index a e I, the mapping ri-»/f(a) of T into F has the corresponding property. We shall be particularly concerned with the case in which F is the field of scalars K, and I is a vector space E over K. Let V be a subspace of KE whose elements are linear forms on E. The topology of simple convergence on such a subspace V is called the weak topology; it is defined by seminorms (12.15.2) />-> !/(*)!, where x runs through E. For each x e E, the mapping/W/(X) is a continuous linear form on V with respect to the weak topology. A sequence (fn) of elementsontained in the convex hull (Problem 13) of/(A).